Theory of complex matrix-valued functions

Question

Let $\mathcal{V}$ be a (complex) vector space and define $S:\mathbb{C}\to M_{n\times n}$ be a complex matrix valued function. In the theory of functions of these types, does there exist analogous results as in the scalar case for theorems such as Cauchys integral formula, Liouville’s theorem etc? I’ve seen results for the case of a function of a matrix, say $f(A)$, but not explicitly matrix-values functions. Any references would be greatly appreciated!